# Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold

Fix $$n\geq 2$$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $$x\in \mathbb{H}^{n}$$ can be represented in polar coordinates $$x=(r, \theta)$$, and equipped with the Riemannian metric $$g=dr^{2}+\sinh^{2}(r)d\theta^{2},$$ where $$dr^{2}$$ is the standard metric on $$\mathbb{R}_{+}$$ and $$d\theta^{2}$$ is the standard metric on the sphere $$\mathbb{S}^{n-1}$$. Denote with $$\mu$$ the Riemannian measure on $$\mathbb{H}^{n}$$ and with $$\sigma$$ the Riemannian measure of co-dimension $$1$$ on hypersurfaces on $$\mathbb{H}^{n}$$. It is a known fact that $$\mathbb{H}^{n}$$ admits the Isoperimetric inequality $$\sigma(\partial \Omega)\geq f(\mu(\Omega)),$$ for all precompact open sets $$\Omega\subset \mathbb{H}^{n}$$ with smooth boundary, where $$f$$ is defined by $$f(v)=c_{1}\left\{\begin{array}{cl}v, &v\geq 1\\ v^{\frac{n-1}{n}}, &v\leq 1,\end{array}\right.$$ for some small constant $$c_{1}>0$$.

Let us fix some precompact open set $$U\subset \mathbb{H}^{n}$$ and consider $$\mathbb{H}^{n}\setminus U$$ as a manifold with boundary $$\partial U$$.

Now my question:

Does the same Isoperimetric inequality now hold for precompact open sets $$\Omega\subset \mathbb{H}^{n}\setminus U$$ with smooth boundary (and possibly a smaller constant $$c_{2}>0$$) and if yes, where can I find a reference for this statement?

Specified question:

Let $$U$$ be a "nice" precompact open set, for example $$U=B_{1}(o)$$ is an open ball of radius $$1$$ for some point $$o\in \mathbb{H}^{n}$$. Does the same Isoperimetric inequality now hold for precompact open sets $$\Omega\subset \mathbb{H}^{n}\setminus B_{1}(o)$$ with smooth boundary (and possibly a smaller constant $$c_{2}>0$$)?

Note that we assume that, when considering a precompact open set $$\Omega\subset \mathbb{H}^{n}\setminus U$$ with smooth boundary, we have $$\partial U\cap \partial \Omega=\emptyset$$.

If this is not known for the hyperbolic space $$\mathbb{H}^{n}$$, is a similar statement known for other Riemannian manifolds, for example $$\mathbb{R}^{n}$$?

• What do you mean by zero Neumann boundary condition in this context? Jul 5, 2021 at 13:59
• Assuming the answer to my first comment is that only the part of $\partial U$ on $\Omega$ is not counted, you cannot have a profile $f$ independent of $\Omega$ unless you impose some condition (bottlenecks could allow small perimeter and large volumes). Such an inequality does hold when $\Omega$ is convex. Jul 5, 2021 at 14:02
• Yes, it is meant as you assume it in your answer. Jul 5, 2021 at 14:33
• Do you have a reference for that statement, when $\Omega$ is convex? Jul 5, 2021 at 14:37
• I specified my question. Jul 5, 2021 at 14:52

Some results are known under some restrictions. First, no uniform inequality can hold if $$U$$ is unrestrained: it could take the shape of (a neighborhood of) a bottle, and the bottleneck will make it possible to have $$\Omega$$ with large volume and small boundary. A natural restriction is to consider convex $$U$$ (so that the double of $$\mathbb{H}^n$$ still is negatively curved in the metric sense).

To sum up: assuming $$U$$ is convex, we even have sharp bounds. When $$M=\mathbb{H}^n$$ the dimensions $$n=2,3,4$$ are covered; when $$M=\mathbb{R}^n$$, you have a weaker inequality (the isoperimetric function has the form $$f(v) = cv^{\frac{n-1}n}$$, there is no linear asymptotic) but known in all dimensions.

Added in edit: I think that an equality $$\sigma(\partial \Omega)\ge c\mu(\Omega)$$ is true in all dimensions whenever $$U$$ is convex, and that the method in our paper with Kuperberg mentionned below can be used to prove this. The more precise bound for small $$v$$, I am not sure.

Note that $$\mathbb{H}^n$$ can even be replaced by a simply connected manifold $$M$$ of sectional curvature bounded above by some $$\kappa\le 0$$. In this setting:

1. Choe proved in 2003 gave the sharp inequality when $$M=\mathbb{R}^n$$ and $$U$$ is a ball, as well as several other restricted cases: Relative isoperimetric inequality for domains outside a convex set. Archives Inequalities Appl 1 (2003): 241-250.

2. In 2007, the general case of a convex set $$U\subset \mathbb{R}^n$$ was obtained by Choe, Ghomi and Ritoré. The relative isoperimetric inequality outside convex domains in $$\mathbf{R}^n$$. Calculus of Variations and Partial Differential Equations 29.4 (2007): 421-429.

3. In 2006, Choe treated the case when $$M$$ has variable (nonpositive) curvature and dimension $$4$$. The double cover relative to a convex domain and the relative isoperimetric inequality. Journal of the Australian Mathematical Society 80.3 (2006): 375-382.

4. In dimension $$3$$, the same was achieved by Choe and Ritoré in 2007. The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds. J. für die reine und angewandte Mathematic (2007): 179-191. They also obtain the sharp inequality when $$\kappa=-1$$, in particular for $$M=\mathbb{H}^3$$.

5. With Kuperberg, we gave a new proof of Choe's result from 2006, treating dimensions $$2$$ and $$4$$ and including (sharp) results when $$M$$ has curvature bounded above by $$\kappa$$ (either $$0$$ or negative). The Cartan–Hadamard conjecture and the Little Prince. Revista Matemática Iberoamericana 35.4 (2019): 1195-1258. Our results can be used to tackle some finite union of convex sets (you need to ensure that geodesic rays reflecting on the boundary of $$U$$ can only bounce a bounded number of time).

I may have missed other relevant references, but you should catch them by looking at papers citing the above ones.

• Thank you very much for your answer! Going through these papers I was wondering if in the case of $\mathbb{H}^{n}$, the profile $f$ is exactly as it I have written it in my Post with $f$ being linear for large volumes? Jul 7, 2021 at 10:28
• @Shaq155: yes, more precisely the profile (over all convex $U$) is $f(v)=\frac12 f_0(2v)$ where $f_0$ is the profile of $\mathbb{H}^n$, which is linear for large $v$. This comes from the optimal case, a half ball against a flat wall. Jul 8, 2021 at 9:34
• @ Benoît Kloeckner: where exactly can I find the method which proves the linear asymptotic for large volumes in all dimensions in your paper with kuperberg or has there been made some progress by now? Dec 4, 2021 at 11:20
• @Shaq155: the relevant result in our paper with Kuperberg is Theorem 1.8; for this sharp bound, it is unfortunately necessary to digest most of the paper, which I admit is somewhat hard. But to prove a mere linear asymptotic, you can adapt the classical trick for the linear inequality in a Cartan-Hadamard manifold with curvature $\le -1$. The trick is to use Stokes equality on the gradient of either a Buseman function of the distance to a point. The adaptation for convex $U$ is to consider the double of $M\setminus U$ along $\partial U$, which is still CAT(-1). Dec 4, 2021 at 12:28